Provability interpretations of modal logic

@article{Solovay1976ProvabilityIO,
  title={Provability interpretations of modal logic},
  author={R. Solovay},
  journal={Israel Journal of Mathematics},
  year={1976},
  volume={25},
  pages={287-304}
}
  • R. Solovay
  • Published 1976
  • Mathematics
  • Israel Journal of Mathematics
We consider interpretations of modal logic in Peano arithmetic (P) determined by an assignment of a sentencev* ofP to each propositional variablev. We put (⊥)*=“0 = 1”, (χ → ψ)* = “χ* → ψ*” and let (□ψ)* be a formalization of “ψ)* is a theorem ofP”. We say that a modal formula, χ, isvalid if ψ* is a theorem ofP in each such interpretation. We provide an axiomitization of the class of valid formulae and prove that this class is recursive. 
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